Book contents
- Frontmatter
- Contents
- Introduction
- 1 Groups and semigroups: connections and contrasts
- 2 Toward the classification of s-arc transitive graphs
- 3 Non-cancellation group computation for some finitely generated nilpotent groups
- 4 Permutation and quasi-permutation representations of the Chevalley groups
- 5 The shape of solvable groups with odd order
- 6 Embedding in finitely presented lattice-ordered groups: explicit presentations for constructions
- 7 A note on abelian subgroups of p-groups
- 8 On kernel flatness
- 9 On proofs in finitely presented groups
- 10 Computing with 4-Engel groups
- 11 On the size of the commutator subgroup in finite groups
- 12 Groups of infinite matrices
- 13 Triply factorised groups and nearrings
- 14 On the space of cyclic trigonal Riemann surfaces of genus 4
- 15 On simple Kn-groups for n = 5, 6
- 16 Products of Sylow subgroups and the solvable radical
- 17 On commutators in groups
- 18 Inequalities for the Baer invariant of finite groups
- 19 Automorphisms with centralizers of small rank
- 20 2-signalizers and normalizers of Sylow 2-subgroups in finite simple groups
- 21 On properties of abnormal and pronormal subgroups in some infinite groups
- 22 P-localizing group extensions
- 23 On the n-covers of exceptional groups of Lie type
- 24 Positively discriminating groups
- 25 Automorphism groups of some chemical graphs
- 26 On c-normal subgroups of some classes of finite groups
- 27 Fong characters and their fields of values
- 28 Arithmetical properties of finite groups
- 29 On prefrattini subgroups of finite groups: a survey
- 30 Frattini extensions and class field theory
- 31 The nilpotency class of groups with fixed point free automorphisms of prime order
26 - On c-normal subgroups of some classes of finite groups
Published online by Cambridge University Press: 20 April 2010
- Frontmatter
- Contents
- Introduction
- 1 Groups and semigroups: connections and contrasts
- 2 Toward the classification of s-arc transitive graphs
- 3 Non-cancellation group computation for some finitely generated nilpotent groups
- 4 Permutation and quasi-permutation representations of the Chevalley groups
- 5 The shape of solvable groups with odd order
- 6 Embedding in finitely presented lattice-ordered groups: explicit presentations for constructions
- 7 A note on abelian subgroups of p-groups
- 8 On kernel flatness
- 9 On proofs in finitely presented groups
- 10 Computing with 4-Engel groups
- 11 On the size of the commutator subgroup in finite groups
- 12 Groups of infinite matrices
- 13 Triply factorised groups and nearrings
- 14 On the space of cyclic trigonal Riemann surfaces of genus 4
- 15 On simple Kn-groups for n = 5, 6
- 16 Products of Sylow subgroups and the solvable radical
- 17 On commutators in groups
- 18 Inequalities for the Baer invariant of finite groups
- 19 Automorphisms with centralizers of small rank
- 20 2-signalizers and normalizers of Sylow 2-subgroups in finite simple groups
- 21 On properties of abnormal and pronormal subgroups in some infinite groups
- 22 P-localizing group extensions
- 23 On the n-covers of exceptional groups of Lie type
- 24 Positively discriminating groups
- 25 Automorphism groups of some chemical graphs
- 26 On c-normal subgroups of some classes of finite groups
- 27 Fong characters and their fields of values
- 28 Arithmetical properties of finite groups
- 29 On prefrattini subgroups of finite groups: a survey
- 30 Frattini extensions and class field theory
- 31 The nilpotency class of groups with fixed point free automorphisms of prime order
Summary
Abstract
A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ HG, where HG ≕ Core(H) is the maximal normal subgroup of G which is contained in H. We obtain the c-normal subgroups in symmetric and dihedral groups. Also we find the number of c-normal subgroups of order 2 in symmetric groups. We conclude by giving a program in GAP for finding c-normal subgroups.
AMS Classification: 20D25.
Keywords: c-normal, symmetric, dihedral.
Introduction
The relationship between the properties of maximal subgroups of a finite group G and the structure of G has been studied extensively. The normality of subgroups in a finite group plays an important role in the study of finite groups. It is well known that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G.
In Wang introduced the concept of c-normality of a finite group. He used the c-normality of a maximal subgroup to give some conditions for the solvability and supersolvability of a finite group. For example, he showed that G is solvable if and only if M is c-normal in G for every maximal subgroup M of G.
In this paper, we obtain the c-normal subgroups in symmetric and dihedral groups, and also we find the number of c-normal subgroups of order 2 in symmetric groups.
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- Groups St Andrews 2005 , pp. 640 - 643Publisher: Cambridge University PressPrint publication year: 2007